Geometry and topology · Maths EE idea · Accessible
The reflection properties of conics
A research question to start from
Why do parabolas, ellipses and hyperbolas reflect rays through their foci, and can all three properties be proved with one method?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Three results unified by one approach: a clear argument with a satisfying conclusion.
Mathematics you would need
- Equations of conics
- Tangents by differentiation
- Angles between lines
- Focus–directrix definitions
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Prove the parabola's property with calculus.
- Prove the ellipse's with the string definition.
- Find a single approach that covers all three (for example the focus–directrix definition).
Scope and difficulty
Accessible. Accessible.
Pitfalls
- Applications (satellite dishes) taking over.
- Different methods with no comparison.
Where to start reading
Search a library catalogue or a university's open lecture notes for: reflection property parabola ellipse hyperbola proof focus directrix. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).